Question 161
Determine all positive real numbers $a$ such that there exists a positive integer $n$ and sets $A_1, A_2, \dots, A_n$ satisfying the following conditions:
(1) every set $A_i$ has infinitely many elements;
(2) every pair of distinct sets $A_i$ and $A_j$ do not share any common element;
(3) the union of sets $A_1, A_2, \dots, A_n$ is the set of all integers;
(4) for every set $A_i$, the positive difference of any pair of elements in $A_i$ is at least $a^i$. (posed by Yuan Hanhui)
(1) every set $A_i$ has infinitely many elements;
(2) every pair of distinct sets $A_i$ and $A_j$ do not share any common element;
(3) the union of sets $A_1, A_2, \dots, A_n$ is the set of all integers;
(4) for every set $A_i$, the positive difference of any pair of elements in $A_i$ is at least $a^i$. (posed by Yuan Hanhui)